Convective diffusion from rising bubbles

Let us estimate the additional pressure drop caused by the presence of surfactant in the liquid. Although in reality the front and rear parts of the bubble s surface have slightly different shapes (the curvature radius of the back cap is greater than that of the front cap ), for simplicity s sake, we consider them as identical hemispheres with radii equal to the radius of the capillary. The length of the liquid film confined between the wall and the bubble is 1. The presence of a surfactant in the liquid flowing around the motionless bubble, leads to a transfer of the surfactant to the bubble surface via convective diffusion. This gives rise to a non-uniform distribution of the surfactant at the bubble surface. The surfactant is pushed to the back of the bubble and accumulates there. An increased surfactant concentration caused a reduction of S. Therefore, 2 decreases from the front to the rear of the bubble. As a result, the pressure in the rear becomes higher than in the front the bubble, and the difference p2 — pi should increase the velocity of the bubble s motion. 

Figure 7-19. A pictorial representation of the distribution of surfactant on the surface of a rising gas bubble. The small sticks at the bubble interface are intended to represent surfactant, which adsorbs preferentially at the gas-liquid interface. The fluid motion, from the top of the bubble toward the bottom, convects surfactant toward the rear of the bubble where it tends to accumulate. This tendency is counteracted to some extent by diffusion that tends toward a uniform surfactant distribution.

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